The Experts below are selected from a list of 234 Experts worldwide ranked by ideXlab platform
V Suresh - One of the best experts on this subject based on the ideXlab platform.
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local global galois theory of Arithmetic Function fields
Israel Journal of Mathematics, 2019Co-Authors: David Harbater, Julia Hartmann, Daniel Krashen, Raman Parimala, V SureshAbstract:We study the relationship between the local and global Galois theory of Function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a Function field. In this context we also study algebraic versions of van Kampen′s theorem, describing the global absolute Galois group as a direct limit of local absolute Galois groups.
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local global galois theory of Arithmetic Function fields
arXiv: Rings and Algebras, 2017Co-Authors: David Harbater, Julia Hartmann, Daniel Krashen, Raman Parimala, V SureshAbstract:We study the relationship between the local and global Galois theory of Function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a Function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups.
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local global principles for zero cycles on homogeneous spaces over Arithmetic Function fields
arXiv: Algebraic Geometry, 2017Co-Authors: Jeanlouis Colliotthelene, David Harbater, Julia Hartmann, Daniel Krashen, Raman Parimala, V SureshAbstract:We study the existence of zero-cycles of degree one on varieties that are defined over a Function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the Function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case.
David Harbater - One of the best experts on this subject based on the ideXlab platform.
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local global galois theory of Arithmetic Function fields
Israel Journal of Mathematics, 2019Co-Authors: David Harbater, Julia Hartmann, Daniel Krashen, Raman Parimala, V SureshAbstract:We study the relationship between the local and global Galois theory of Function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a Function field. In this context we also study algebraic versions of van Kampen′s theorem, describing the global absolute Galois group as a direct limit of local absolute Galois groups.
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local global galois theory of Arithmetic Function fields
arXiv: Rings and Algebras, 2017Co-Authors: David Harbater, Julia Hartmann, Daniel Krashen, Raman Parimala, V SureshAbstract:We study the relationship between the local and global Galois theory of Function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a Function field. In this context we also study algebraic versions of van Kampen's theorem, describing the global absolute Galois group as a pushout of local absolute Galois groups.
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local global principles for zero cycles on homogeneous spaces over Arithmetic Function fields
arXiv: Algebraic Geometry, 2017Co-Authors: Jeanlouis Colliotthelene, David Harbater, Julia Hartmann, Daniel Krashen, Raman Parimala, V SureshAbstract:We study the existence of zero-cycles of degree one on varieties that are defined over a Function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the Function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case.
Liu Zhe - One of the best experts on this subject based on the ideXlab platform.
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A Family of Lightweight Twisted Edwards Curves for the Internet of Things
Springer International Publishing, 2019Co-Authors: Ghatpande Sankalp, Großschädl Johann, Liu ZheAbstract:Part 6: Internet of ThingsInternational audienceWe introduce a set of four twisted Edwards curves that satisfy common security requirements and allow for fast implementations of scalar multiplication on 8, 16, and 32-bit processors. Our curves are defined by an equation of the form $$-x^2 + y^2 = 1 + dx^2y^2$$ over a prime field $$\mathbb {F}_p$$, where d is a small non-square modulo p. The underlying prime fields are based on “pseudo-Mersenne” primes given by $$p = 2^k - c$$ and have in common that $$p \equiv 5 \bmod {8}$$, k is a multiple of 32 minus 1, and c is at most eight bits long. Due to these common features, our primes facilitate a parameterized implementation of the low-level Arithmetic so that one and the same Arithmetic Function is able to process operands of different length. Each of the twisted Edwards curves we introduce in this paper is birationally equivalent to a Montgomery curve of the form $$-(A+2)y^2 = x^3 + Ax^2 + x$$ where $$4/(A+2)$$ is small. Even though this contrasts with the usual practice of choosing A such that $$(A + 2)/4$$ is small, we show that the Montgomery form of our curves allows for an equally efficient implementation of point doubling as Curve25519. The four curves we put forward roughly match the common security levels of 80, 96, 112 and 128 bits. In addition, their Weierstraß representations are isomorphic to curves of the form $$y^2 = x^3 - 3x + b$$ so as to facilitate inter-operability with TinyECC and other legacy software
Zhe Liu - One of the best experts on this subject based on the ideXlab platform.
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A Family of Lightweight Twisted Edwards Curves for the Internet of Things
2018Co-Authors: Sankalp Ghatpande, Johann Großschädl, Zhe LiuAbstract:We introduce a set of four twisted Edwards curves that satisfy common security requirements and allow for fast implementations of scalar multiplication on 8, 16, and 32-bit processors. Our curves are defined by an equation of the form $$-x^2 + y^2 = 1 + dx^2y^2$$ over a prime field $$\mathbb {F}_p$$, where d is a small non-square modulo p. The underlying prime fields are based on “pseudo-Mersenne” primes given by $$p = 2^k - c$$ and have in common that $$p \equiv 5 \bmod {8}$$, k is a multiple of 32 minus 1, and c is at most eight bits long. Due to these common features, our primes facilitate a parameterized implementation of the low-level Arithmetic so that one and the same Arithmetic Function is able to process operands of different length. Each of the twisted Edwards curves we introduce in this paper is birationally equivalent to a Montgomery curve of the form $$-(A+2)y^2 = x^3 + Ax^2 + x$$ where $$4/(A+2)$$ is small. Even though this contrasts with the usual practice of choosing A such that $$(A + 2)/4$$ is small, we show that the Montgomery form of our curves allows for an equally efficient implementation of point doubling as Curve25519. The four curves we put forward roughly match the common security levels of 80, 96, 112 and 128 bits. In addition, their Weierstraß representations are isomorphic to curves of the form $$y^2 = x^3 - 3x + b$$ so as to facilitate inter-operability with TinyECC and other legacy software.
Victor Volfson - One of the best experts on this subject based on the ideXlab platform.
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Summation Arithmetic Functions with bounded terms, having a limit normal distribution law.
arXiv: Number Theory, 2019Co-Authors: Victor VolfsonAbstract:The paper considers the properties of pseudo stationarity in a broad sense and pseudo strong mixing for sequences of random variables corresponding to Arithmetic Functions. Assertions on this topic have been proven. The implementation of these properties for known Arithmetic Functions has been verified. The article proves a statement about sufficient conditions under which a summation Arithmetic Function with bounded terms has a limit normal distribution law. The fulfillment of the specified sufficient conditions for known Arithmetic Functions is considered.
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Summation Arithmetic Functions with asymptotically independent summands
arXiv: Number Theory, 2018Co-Authors: Victor VolfsonAbstract:The summation Arithmetic Functions with asymptotically independent summands are studied in the paper. We prove statements about the condition under which the summation Arithmetic Functions have asymptotically independent summands. It is also prove that the limiting distribution of the summation Arithmetic Function with asymptotically independent summands is normal under certain conditions for summands of the summation Arithmetic Function.
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Investigations of the limit distribution and the asymptotic behavior of summation Arithmetic Functions
arXiv: Number Theory, 2018Co-Authors: Victor VolfsonAbstract:The aim of the paper is to study the limit distributions and the asymptotic behavior of summation Arithmetic Functions. A probabilistic approach based on the use of the axioms of probability theory is used for these purposes. Sufficient conditions are proved under which these Functions have a limiting normal distribution. Arithmetic Functions having a limiting normal distribution are found in the paper. The author investigates summation Functions of a general form and finds sufficient conditions under which they have a limiting normal distribution. Examples of Arithmetic Functions that satisfy these requirements are considered. We prove an ergodic theorem for summation Arithmetic Functions, for which the sequence of random variables has the stationarity property in the broad sense. The asymptotics of the growth of the deviation of the values of the Arithmetic Function from its mean value are investigated. It is shown that an equivalent formulation of the Riemann hypothesis for the Mertens Function is satisfied almost everywhere.